Méthodes numériques en ingénierie financière

Objectif

Participants of this course will master computational techniques frequently used in mathematical finance applications.

Prerequisites: Stochastic processes / stochastic calculus, Numerical Analysis, Derivatives, Command of Python

TDs: In the form of ipython notebooks

Plan

1. Brief introduction to asset price modeling and option pricing

• Basic stochastic models in finance
• Introduction to numerical methods for option pricing

2.  Monte Carlo methods for pricing

• Euler schemes and applications to volatility models
• Variance reduction 

3. Option pricing via PDEs

• Finite difference approximation of Black-Scholes PDE
• American options and free boundary problems

4. Transformation based methods

• Affine models
• Option pricing via Fourier transforms

5. Density approximation techniques

• Polynomial models and calculation of moments
• Option pricing via density approximation

6. Rough Volatility

• Motivation for rough volatility models and examples
• The rough Heston model 

7. Machine learning methods for pricing

• Parametric option pricing and calibration with Neural Networks
• Neural networks for solving PDEs

Références

·      Achdou, Yves; Pironneau, Olivier. Computational methods for option pricing. Frontiers in Applied Mathematics, 30. SIAM, Philadelphia, PA, 2005.

·      Björk, Tomas. Arbitrage theory in continuous time. Third edition, OUP Oxford, 2009.

·      Gatheral,  Jim. Volatility Surface: A Practitioner’s Guide. Wiley, 2006.

·      Glasserman, Paul. Monte Carlo methods in financial engineering. Springer, 2003.

·      Hilber, Norbert; Reichmann, Oleg; Schwab, Christoph; Winter, Christoph. Computational methods for quantitative finance. Springer, 2013.

·      Hirsa, Ali. Computational methods in finance. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton, FL, 2013.

·      Lamberton, Damien; Lapeyre, Bernard. Introduction to stochastic calculus applied to finance. Second revised edition. Chapman & Hall/CRC, 2008.

·      Seydel, Rüdiger U. Tools for computational finance. Fourth edition. Universitext. Springer-Verlag, Berlin, 2009.

·      Shreve, Steven E. Stochastic calculus for finance II: Continuous-Time models, Volume 11. Springer Science & Business Media, 2004.

·      Additional lecture material will be provided by the instructors.